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A regular hexagon has a perimeter of $p$ (in length units) and an area of $A$ (in square units). If $A = \frac{3}{2},$ then find the side length of the hexagon.

 Mar 15, 2024
 #1
avatar+128406 
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The side of the hexagon =  (1/6)p

 

1/6 of the  area is a triangle with sides of (1/6)p and an included angle of 60°

 

So

 

(1/6) (3/2)  =   (1/2) [(1/6)p]^2 (sqrt (3) / 2)

 

1/4  = (1/4) (sqrt (3)) / 36 p^2

 

1 = [ sqrt 3] / 36 *  p^2

 

p^2  = 36 / sqrt (3)

 

p = 6 / 3^(1/4)

 

side =  (1/6) * 6 / 3^(1/4)  = 1 / 3^(1/4)  ≈  .76

 

cool cool cool

 Mar 15, 2024

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